An introduction to the modern theory of equations, by Florian Cajori.

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THEORY OF EQUATIONS

In the third step, f' = 4 =- e'f" = 2 ~ 2,
71 = c +,016, it4 = W13 + 04,
7/11 =0 3 + 014, lt5 = 05 + 012
~/12 =- 9 f i8,,i6 = 015 + 2,
7 i3 =_ 10 + W7, W17I = 7 11 + 6.
Since nt" and *1"4 have A' for their sum and?/l for their product, they are
the roots of
x2 - nIX + 71=- 0,
and we obtain ~ "1 = + -i
2 _4
Finally we find that co and w16 are roots of the quadratic
x2 _ xix _ 1 = 0;
that is, wO i_ - + ' 1 _,2
a primitive root of the cyclotomic equation of degree 16.
After solving one of the quadratics given above, the question arises,
which one of the two roots represents a given period? For instance,
which of the roots of x - tix - = 0 represents v'l? To settle this,
form the product
(' - '2)(X/'1 - 3) = 2(77 - )-+1) - N=4 V1 + 1( ' - ').
Hence Yl _- q73 is positive, and lI/ has the plus sign before its radical,
773 the negative sign.
It is readily seen that, since the equation x17 -1 = 0 involves in its
solution no other irrationals than square roots, a regular polygon of seventeen sides can be inscribed in a circle by means of the ruler and compasses.
Gauss discovered a method of inscribing this polygon when he was a youth
of nineteen years. It was this discovery which induced him to pursue
mathematics as his life-work rather than languages. For an explanation
of the construction of the regular seventeen-sided polygon consult Bachmann, Lehre von der Kreistheilung, Leipzig, 1872, p. 67, or Klein's Famous
Problems of Elementary Geometry (ed. W. W. Beman and D. E. Smith),
Boston, 1897, p. 41. We have followed Bachmann's exposition of the
subject of the division of the circle.